Lie algebras and the are key concepts in algebraic groups. They provide a powerful tool for studying the local structure of Lie groups, connecting abstract algebra with geometry and analysis.
The exponential map bridges Lie algebras and Lie groups, allowing us to construct one-parameter subgroups and explore group properties. It's essential for understanding the structure and classification of algebraic groups, including simple and semisimple Lie algebras.
Lie algebras and algebraic groups
Definition and properties of Lie algebras
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A is a vector space g over a field F equipped with a binary operation called the , denoted [⋅,⋅], satisfying:
Bilinearity: [aX+bY,Z]=a[X,Z]+b[Y,Z] and [X,aY+bZ]=a[X,Y]+b[X,Z] for a,b∈F and X,Y,Z∈g
Alternativity: [X,X]=0 for all X∈g
: [X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0 for all X,Y,Z∈g
Examples of Lie algebras include:
The vector space of n×n matrices over a field F with the commutator bracket [A,B]=AB−BA (gln(F))
The vector space of smooth vector fields on a manifold with the Lie bracket of vector fields (X(M))
Relation between Lie algebras and algebraic groups
The Lie algebra g of an algebraic group G is the tangent space at the identity element, g=TeG, with the Lie bracket given by the commutator of left-invariant vector fields
For each φ:G→H, there is an associated Lie algebra homomorphism dφ:g→h between their corresponding Lie algebras, preserving the Lie bracket structure
The of a Lie group G on its Lie algebra g is given by the differential of the conjugation map, Ad:G→GL(g), where Ad(g)(X)=d(Cg)e(X) for g∈G and X∈g, and Cg(h)=ghg−1 is the conjugation map
Examples of the correspondence between Lie groups and Lie algebras:
The Lie algebra of the general linear group GL(n,C) is the space of n×n complex matrices gln(C)
The Lie algebra of the special orthogonal group SO(n) is the space of skew-symmetric n×n real matrices son
Exponential map for Lie groups
Definition and properties of the exponential map
The exponential map exp:g→G is a local diffeomorphism from a neighborhood of 0 in the Lie algebra g to a neighborhood of the identity element in the Lie group G
For a matrix Lie group G⊆GL(n,C), the exponential map is given by the matrix exponential: exp(X)=∑k=0∞k!Xk for X∈g
The exponential map satisfies the property exp(sX)exp(tX)=exp((s+t)X) for s,t∈R and X∈g, making it a group homomorphism from (g,+) to (G,⋅)
The derivative of the exponential map at 0∈g is the identity map, d(exp)0=idg
Examples of exponential maps:
For the Lie group GL(n,C), the exponential map is the matrix exponential exp(A)=∑k=0∞k!Ak for A∈gln(C)
For the Lie group SO(3), the exponential map is given by the Rodrigues' rotation formula exp(θn^)=I+sinθ[n^]×+(1−cosθ)[n^]×2, where θn^∈so3 is a skew-symmetric matrix representing a rotation by angle θ about the axis n^
Construction of one-parameter subgroups
The exponential map is used to construct one-parameter subgroups of a Lie group, which are smooth homomorphisms from (R,+) to (G,⋅) of the form t↦exp(tX) for some X∈g
One-parameter subgroups are the integral curves of left-invariant vector fields on the Lie group G
The exponential map establishes a bijection between the Lie algebra g and the set of one-parameter subgroups of G
Examples of one-parameter subgroups:
In the Lie group GL(n,C), the one-parameter subgroup generated by a matrix A∈gln(C) is given by t↦exp(tA)=∑k=0∞k!tkAk
In the Lie group SO(3), the one-parameter subgroup generated by a skew-symmetric matrix θn^∈so3 represents rotations about the axis n^ by angles proportional to t
Properties and applications of the exponential map
Surjectivity and generating properties
The exponential map is surjective for connected, simply connected Lie groups, meaning every element of G can be written as exp(X) for some X∈g
For compact Lie groups, the exponential map is not necessarily surjective, but its image generates a dense subset of the group
Examples:
The exponential map of the Lie group SU(2), the group of 2×2 unitary matrices with determinant 1, is surjective because SU(2) is simply connected
The exponential map of the Lie group SO(3), the group of 3×3 orthogonal matrices with determinant 1, is not surjective, but its image generates a dense subset of SO(3)
Baker-Campbell-Hausdorff formula
The Baker-Campbell-Hausdorff formula expresses the product of two exponentials in a Lie group in terms of a single exponential: exp(X)exp(Y)=exp(Z), where Z is a series in terms of nested Lie brackets of X and Y
The first few terms of the Baker-Campbell-Hausdorff formula are: Z=X+Y+21[X,Y]+121[X,[X,Y]]−121[Y,[X,Y]]+...
The Baker-Campbell-Hausdorff formula is useful for approximating products in Lie groups and studying the local structure of Lie groups near the identity
Example: In the Lie group GL(n,C), the Baker-Campbell-Hausdorff formula gives an approximation of the product of two matrices close to the identity: exp(A)exp(B)≈exp(A+B+21[A,B]) for small matrices A,B∈gln(C)
Structure and classification of Lie algebras
Simple and semisimple Lie algebras
A Lie algebra is simple if it has no non-trivial ideals and is not abelian. The classification of simple Lie algebras over C leads to the four infinite families An,Bn,Cn,Dn, and five exceptional Lie algebras G2,F4,E6,E7,E8
A Lie algebra is semisimple if it is a direct sum of simple Lie algebras. The , a symmetric bilinear form on g, is non-degenerate if and only if g is semisimple
Examples of simple and semisimple Lie algebras:
The Lie algebra sln(C) of traceless n×n complex matrices is simple for n≥2
The Lie algebra son(C) of skew-symmetric n×n complex matrices is simple for n≥5 and semisimple for n=3,4
Levi decomposition and solvable radical
The Levi decomposition states that any g can be written as a semidirect product g=s⋉r, where s is a semisimple subalgebra (the Levi factor) and r is the solvable radical
The solvable radical r is the maximal solvable ideal of g, and it consists of the nilpotent and abelian parts of the Lie algebra
The Levi decomposition is unique up to conjugation by elements of the group exp(r)
Example: The Lie algebra of the Poincaré group, the semidirect product of translations and Lorentz transformations in special relativity, has a Levi decomposition p=so(1,3)⋉R4, where so(1,3) is the Lorentz Lie algebra and R4 is the of translations
Root systems and classification
The structure of a is determined by its root system, a configuration of vectors in a Euclidean space satisfying certain integrality and reflection conditions
A root system is a finite set Φ of non-zero vectors in a Euclidean space 𝔢 satisfying:
If α ∈ Φ, then the only multiples of α in Φ are ±α
For each α ∈ Φ, the reflection σ_α(β) = β - 2\frac{(β,α)}{(α,α)}α leaves Φ invariant
For all α, β ∈ Φ, the number 2\frac{(β,α)}{(α,α)} is an integer
The classification of root systems leads to the classification of semisimple Lie algebras, with each simple Lie algebra corresponding to an irreducible root system
Examples of root systems and their corresponding Lie algebras:
The root system An corresponds to the special linear Lie algebra sln+1(C)
The root system Cn corresponds to the symplectic Lie algebra sp2n(C)
Key Terms to Review (18)
Abelian lie algebra: An abelian Lie algebra is a type of Lie algebra where the Lie bracket operation is commutative, meaning that the bracket of any two elements is zero. This property signifies that all elements of the algebra can be simultaneously diagonalized, which makes the structure particularly simple and well-behaved. Abelian Lie algebras are fundamental in the study of Lie groups and serve as building blocks for more complex algebras, especially when examining the exponential map and its properties.
Adjoint Representation: The adjoint representation is a way of representing a Lie algebra through its own structure, where each element of the Lie algebra acts as a linear transformation on itself via the Lie bracket. This representation reveals the internal symmetries of the algebra and connects closely with concepts like the exponential map, which relates elements of the Lie algebra to transformations in the associated Lie group.
Closed under lie bracket: A subset of a Lie algebra is said to be closed under the Lie bracket if, for any two elements in that subset, their Lie bracket also belongs to the same subset. This property is essential for ensuring that the subset itself forms a Lie subalgebra, preserving the structure and operations defined on the larger algebra.
Exponential Map: The exponential map is a mathematical function that relates the tangent space at a point on a manifold to the manifold itself, particularly in the context of Lie groups and Lie algebras. It provides a way to 'exponentiate' elements of a Lie algebra to obtain corresponding elements in a Lie group, which is crucial for understanding the relationship between algebraic structures and geometric properties.
Finite-dimensional lie algebra: A finite-dimensional Lie algebra is a vector space equipped with a binary operation called the Lie bracket, satisfying bilinearity, antisymmetry, and the Jacobi identity, where the dimension of the vector space is finite. This structure is crucial for understanding symmetries in mathematics and physics and plays a significant role in the study of algebraic groups and representation theory.
Homomorphism: A homomorphism is a structure-preserving map between two algebraic structures, such as groups, rings, or algebras, that respects the operations defined on those structures. In the context of Lie algebras and the exponential map, a homomorphism helps to relate different algebraic entities and allows us to transfer properties and operations from one algebraic structure to another, which is crucial for understanding their relationships and behaviors.
Irreducible Representation: An irreducible representation is a representation of a group or algebra that cannot be decomposed into smaller, simpler representations. In other words, it is a representation that has no proper invariant subspaces other than the trivial subspace. Understanding irreducible representations is crucial in the study of symmetries and helps in the classification of representations, especially when dealing with Lie algebras and their exponential maps, as well as in analyzing characters in representation theory.
Jacobi Identity: The Jacobi Identity is a fundamental property of Lie algebras, which states that for any elements $x$, $y$, and $z$ in a Lie algebra, the equation $[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$ holds. This identity ensures the consistency of the Lie bracket operation and reflects the anti-symmetry and bilinearity of the bracket in Lie algebras.
Killing form: The Killing form is a bilinear form defined on a Lie algebra, used to measure the 'size' of the Lie algebra in terms of its structure constants. It is a crucial tool for determining properties such as semisimplicity and whether a given Lie algebra is reductive. By analyzing the Killing form, one can deduce important information about the representation theory and the roots of the algebra, linking it to concepts like the exponential map.
Lie Algebra: A Lie algebra is a mathematical structure that consists of a vector space equipped with a binary operation called the Lie bracket, which satisfies bilinearity, antisymmetry, and the Jacobi identity. Lie algebras play a crucial role in various areas of mathematics and physics, particularly in the study of symmetry and the behavior of linear transformations. They provide a framework for understanding the relationship between algebraic structures and geometric objects, often through the exponential map that connects the algebra to Lie groups.
Lie Bracket: The Lie bracket is a binary operation defined on a Lie algebra that captures the essence of the algebra's structure and provides a way to understand its properties. It is often denoted by `[x, y]`, where `x` and `y` are elements of the Lie algebra, and it is antisymmetric, meaning that `[x, y] = -[y, x]`, and satisfies the Jacobi identity, which relates the brackets of three elements. This operation plays a crucial role in studying the relationships between elements of the Lie algebra and in connecting algebraic structures to geometric notions through the exponential map.
Lie group: A Lie group is a mathematical structure that combines the properties of both a group and a smooth manifold, allowing for the study of continuous transformations. It provides a framework for understanding symmetry and smooth transformations in various areas of mathematics and physics, enabling the connection between algebraic operations and geometric interpretations. Lie groups are essential in formulating theories in physics, particularly in areas like gauge theory and quantum mechanics.
Lie's Theorem: Lie's Theorem states that the exponential map is a local diffeomorphism from a Lie algebra to its corresponding Lie group, meaning that small changes in the Lie algebra lead to small changes in the Lie group. This theorem is crucial as it connects algebraic structures (the Lie algebra) with geometric structures (the Lie group), allowing one to use algebraic techniques to study differential equations and geometric properties.
Matrix Lie Algebra: A matrix Lie algebra is a mathematical structure formed by the set of all matrices of a fixed size, where the operations of addition and scalar multiplication are defined, along with a Lie bracket defined as the commutator of two matrices. This concept connects to the broader study of Lie algebras and the exponential map, which are used to analyze continuous symmetry and transformations within various mathematical and physical contexts.
Representation Theory: Representation theory studies how algebraic structures, particularly groups and algebras, can be represented through linear transformations of vector spaces. It connects abstract algebraic concepts to more concrete linear algebra, allowing mathematicians to analyze symmetries and group actions in various mathematical contexts.
Semisimple Lie Algebra: A semisimple Lie algebra is a type of Lie algebra that can be decomposed into a direct sum of simple Lie algebras, which means it has no non-trivial solvable ideals. These algebras are important in the study of symmetries in mathematics and physics, as they play a key role in the classification of finite-dimensional representations and the structure theory of Lie algebras. Semisimple Lie algebras are closely connected to concepts such as the exponential map, which is used to relate Lie algebras to Lie groups.
Sophus Lie: Sophus Lie was a Norwegian mathematician known for founding the theory of continuous transformation groups, now known as Lie groups. His work laid the foundation for the study of symmetry in mathematical structures and has deep connections to various areas, including algebra, geometry, and physics. The concepts he introduced have critical implications in understanding Lie algebras and the exponential map.
Wilhelm Killing: Wilhelm Killing was a German mathematician known for his foundational work in the theory of Lie algebras and their representations. His contributions significantly advanced the understanding of semisimple Lie algebras and the Killing form, which is a crucial bilinear form associated with these algebras, playing an essential role in the structure theory of Lie groups and their corresponding algebraic structures.